110 XII! INTEGRATION (13.4.4) Let (/zn) be an increasing sequence of real measures on X such that, for each function /^ 0 belonging to JfR(X), the sequence (/-*„(/)) is bounded above in R. Then the sequence (X,) has a vague limit in MR(X) which is also its least upper bound for the order-relation on MR(X). For each function /g: 0 in JTR(X), the sequence (jun(/)) is increasing and bounded above in R, hence (4.2.1 ) has a limit //(/) in R which is equal to sup jun(/). Since every function belonging to Jf C(X) is a linear combination n of four functions ^0 belonging to «#* R(X), it follows that the sequence (/in) is vaguely convergent (13.4.1), and it is clear that \i is its least upper bound, by the definition of the order-relation on MR(X). (13.4.5) If a series of positive measures on X, with general term JUB, is such that, for each f*z. 0 in jfR(X), the series with general term nn(f) «£ 0 is con- vergent in R, then the series with general term \in is vaguely convergent in MR(X), and its sum /i = £ /*„ is such that u(f) = £ Hn(f)for allfe Jf R(X). n n Apply (13.4.4) to the partial sums of the series. PROBLEMS 1. Let A be Lebesgue measure on R, let JLG be its restriction to R* = ]0, + °o [, and let g be the function x i—>• l/x on R* . Show that the measure g • jit cannot be extended to a measure on R (cf. (17.9)). 2. On the real line R, show that the sequence of Dirac measures £„ (unit mass at the point +TI) converges vaguely to 0. Give an example of a sequence (/,) in «2fc(R) which converges to 0 in the Frechet space ^C(R) (12.14.6) but is such that the sequence «£n>/n» tends to 1. Let X be a compact space. (a) Show that if (/„) is a sequence of functions in the Banach space ^C(X) which tends to 0 and if (ju.n) is a sequence of measures on X which tends vaguely to 0, then the sequence «/xn,/n» tends to 0 (use (13.4.2)). (b) Suppose that X is infinite. Let V be a neighborhood of 0 in M(X) for the vague topology, defined by a finite number, of inequalities | </*,/}>! ^ 1 (1 ^y ^ m)t and let /be any function belonging to #C(X) which is not a linear combination of the/}. Show that there exists ft e V such that | </x,/>| is arbitrarily large (cf. Section 12.15, Problem 1). Deduce that the mapping (/A,/)i-> </*,/> of #c(X) x M(X) into C is not continuous, and hence (using (a)) that the space M(X) is not metrizable with respect to the vague topology.ch fi e H and eachfe «>f C(X), we have